Macromolecules 1987, 20, 1400-1402
1400
(6) MacKnight, W. J.; Earnest, T. R. J. Polym. Sei.-Macromol. Reu. 1981, 16, 41. (7) Peiffer, D. G.; Hager, B. L.; Weiss, R. A.; Agarwal, P. K.; Lundberg, R. D. J. Polym. Sci., Polym. Phys. Ed. 1985, 23, 1869. (8) Lantman, C. W.; MacKnight, W. J.; Peiffer, D. G.; Sinha, S. K.; Lundberg, R. D. Polym. Prepr. (Am. Chem. Soc., Diu. PoEym. Chem.) 1986, 27(1), 327. (9) Lantman, C. W.; MacKnight, W. J.; Peiffer, D. G.; Sinha, S. K.; Lundberg, R. D. Polym. Prepr. (Am. Chem. Soc., Diu. Polym. Chem.) 1986, 27(2), 292.
(10) Lantman, C. W.; MacKnight, W. J.; Peiffer, D. G.; Sinha, S. K.; Lundberg, R. D. Macromolecules 1987,20, 1096. (11) Lantman, C. W.; MacKnight, W. J.; Higgins, J. S.; Peiffer, D. G.; Sinha, S. K.; Lundberg, R. D. Macromolecules, submitted. (12) Lundberg, R. D.; Makowski, H. S.; Westerman, L. Ions in Polymers; American Chemical Society: Washington, D.C., 1980; Adu. Chem. Ser. 187, p 67. (13) Peiffer, D. G.; Duvdevani, I.; Lundberg, R. D. Polymer 1986, 27. 1453. (14) Lundberg, R. D.; Phillips, R. R., J. Polym. Sci., Polym. Lett. Ed. 1984, 22, 317.
Osmotic Second Virial Coefficient and Two-Parameter Theories Klaus Huber*+*and Walter H. Stockmayed Department of Chemistry, Dartmouth College, Hanover, New Hampshire 03755, and Znstitut fur Makromolekulare Chemie der Universitat Freiburg, 0-7800 Freiburg, FRG. Received December 15, 1986
ABSTRACT: Literature data on polystyrene in toluene show that the molecular weight dependence of the interpenetration function, \k, does not follow the dependence on chain expansion factor, a,,that would be predicted by two-parameter theory for Gaussian chains; there is an increase of at low molecular weight which can be explained by the non-Gaussian dependence of chain dimensions on M caused by local chain stiffness. A different kind of deviation from two-parameter theory is seen under 0-solvent conditions: for polystyrene in cyclohexane at the 0-temperature,second virial coefficients at low M do not vanish and can be rationalized as due to three-body interactions within the overlap region of two polymers.
Small Molecular Weights in Good Solvents Interactions in dilute solutions of flexible high polymers are often discussed in terms of the interpenetration function,' \k. This dimensionless quantity is defined in relation to the osmotic second virial coefficient, Az (for two identical chains), by where M is the polymer molar weight, ( S z ) the mean square radius of gyration, and NA Avogadro's number. Several approximate expressions2 have been derived for 9 as a function of the excluded volume parameter.
Here p is the binary cluster integral and n the number of statistically independent segments of effective bond length, a. The so-called two-parameter theories3 based on the Gaussian chain model involve the only combinations naz and n2@,where n is proportional to M . It is well-known that z cannot be determined directly by experiment. However, the experimentally accessible expansion coefficient, as,for the rms radius of gyration can also be expressed as a function of z. Therefore, a plot of \k(z) vs. a&) eliminates z and provides a valuable test for two-parameter theories. As is the case for \ k ( z ) , several theoretical approximations are available4-' for a, as a function of z. It is important to find self-consistent pairs of functions \k(z) and a,(z). Though several papers&13deal with this test of the two-parameter theories, it appears helpful to add further comments. As one example, the treatments according to Kurata et al.,14 Yamakawa,15and Yamakawa and Tanaka16 give the relations t Dartmouth
*
College. Institut fur Makromolekulare Chemie der Universitat Freiburg.
0024-9297/87/2220-1400$01.50/0
+ 0.459(1 + 6 . 0 4 ~ ) O . ~ ~ ( 3 4 \k = 0.547[1 - (1 + 3 . 9 0 3 ~ / ( ~ , ~ ) ~ '(3b) ~~~] a,' = 0.541
These equations, like other combinations, yield reasonable agreement with many experimental resultsgJObut only under certain circumstances, as will be seen below. We do not suggest that they should necessarily be preferred to other such combinations. Apart from numerical differences, all but ~ w oof ~the~ ~ , ~ approximate theories of \k yield monotonically increasing functions of z. In the limit of high z, they attain constant values. This behavior has been confirmed only by experiments with relatively high molecular weight samples, in which the excluded volume parameter, z(a,),was varied by changing the t e m p e r a t ~ r e . ~ JAn ~ Jexample, ~ Figure 1represents such measurements by Berryla with different polystyrene (PS) samples in trans-decalin. In good solvents, a limiting plateau value is also found experimentally for high molecular weights.lSz3 According to eq 3b, the limiting value is 0.547, which is significantly larger than any experimental data, available in the high z range. However, more recent theoretical predictions, 0.231 by Oono and Freedz4 and 0.268 by Witten and Schaferz5,are in closer agreement with experiment. The crucial point here is that the shape of @(a,) is completely different from that of Figure 1. This is seen in Figure 2, which represents \k isothermally over a wide range of a : for PS in the good solvent toluene and employs several sets of literature data.21-z3,26 In this case, z is varied at constant p by changing the molecular weight. This behavior has already been mentioned by Miyakilg et al. We offer further evidence from small-angle neutron scattering experiment^^^,^^ with four very low molecular weight PS samples in toluene-d8 and cyclohexane-d,,. These provide us with the complete set of data necessary to calculate \k and a, essentially down to the limit a, 1. For the other literature data,22,23,26 ( S2)values for unperturbed PS were calculated with the equation of Benoit
-
0 1987 American Chemical Society
Osmotic Second Virial Coefficient 1401
Macromolecules, Vol. 20, No. 6,1987
01 " Id
zN
IO'
Olo5
Id
L
M
I
2
4
a2
Figure 1. Plot of e vs. a: from Berry's datal8 for PS in trans-decalin. All samples have molecular weights larger than 5 X IO5. The dashed curve is calculated with eq 3a and 3b.
I
2
4
3
4
Figure 3. Second virial coefficient of PS in toluene ( 0 )ref 27 and (0)ref 29. Dashed curves: theory of Yamakawa and Stockmayer" with 0 = 1.6 nm (1)and with = 5.2 nm (2). The drawn curve represents experimental results according to ref 27. Second virial coefficient in cyclohexane at T = 35 O C : @ with error bars.
5
Figure 2. Literature data of vs. a: for PS in toluene: ( 0 )ref 21, (A)ref 22, ( 0 ) ref 26,and (0) ref 33. The expansion factor a,is modified by changing the molecular weight. The drawn line serves as a guide for the eye. and Doty%for wormlike chains, with 2.9 nm as the Kuhn length and 450 g/(mol nm) as the mass per unit length.21 In order t o give a qualitative interpretation of the behavior of we now consider the molecular weight dependence of the second virial coefficient of PS in toluene. If molecular weights are lower than about lo4, the experimentally determined expansion coefficient, a,,approaches unity.27 The two-parameter theories, being based on Gaussian chains, require that when a, = 1,the quantities z, and A2 should approach 0. However, the experimental behavior is quite different, as is shown in Figure 3, which again represents literature data27v29for PS in toluene. A proper comparison between theory and experiment a t low M should be based on non-Gaussian models. Here we make use of the excluded volume calculations by Yamakawa and Stockmayer30 for wormlike chains. These authors calculated the effects of chain stiffness on the single contact term in the cluster series for the expansion factor and on the double-contact term in the series for A2, and they assumed that the effect of stiffness on higher terms could be scaled according to the number of contacts. The result is that Kz replaces 1.3332 in the equation for a, and Qz replaces 2.8662 in the equation for iP, where K and Q are functions of contour length available from Table I and eq 96 and 119 of ref 30. Applying this scheme to eq 3, we now have
*,
*
+ 0.459[1 + 4.53K~lO.~~ (3a') iP = 0.638Q[1 + 1 . 3 6 2 3 Q ( ~ / a , 3 ) - ~ . ~ ~(3b') ] a, = 0.541
The excluded volume parameter, 2, was calculated by eq
4
8
12
Np
16
20
Figure 4. Literature data for e plotted against Nk1I2 for PS in toluene; symbols as in Figure 2;the dashed curves correspond to those of Figure 3. Here Nk = n is the number of Kuhn segments. 2, where a and n become the Kuhn length and the number of Kuhn segments, respectively, both estimated according to ref 21. The binary cluster integral, @,was treated as a fitting parameter. Calculations were carried out with two different values for p. The results of the procedure just described are represented in Figures 3 and 4. Whereas the value @ = 1.6 nm3 provides good agreement in the high molecular weight range, a considerably higher value of @ = 5.2 nm3 has to be assumed in order to achieve agreement in the low molecular weight range. However, the main feature of the calculations is an increase of \k with decreasing molecular weight in both cases (Figure 4). This can only be achieved by applying a theory which takes chain stiffness into account. In the low molecular weight range, stiffness causes an increase of the exponent of ( S2)toward the value for rods. At the same time, A2 should assume a plateau value.31 Therefore, we must have the following limiting behavior \k
-
-
A2W/(S2)3/2 M-' (low M, good solvent)
(4)
It thus appears that the interpenetration function is of only limited use in good solvent systems. Small Molecular Weights under @Conditions Light scattering experiments were performed on four low-M PS samples in cyclohexane a t 35 "C. All experiments were carried out on a Sofica light-scattering apparatus a t a wavelength of 436 nm. For each sample, a t least eight different concentrations were measured. Prior to use,
1402 Huber and Stockmayer
Macromolecules, Vol. 20, No. 6,1987
cyclohexane was distilled over Na wire. To remove dust, each solution was filtered through a Millipore filter (0.05 pm). No angular dependence could be detected, and an average value of Kc/Rewas calculated from measurements a t angles of 60°, 90°, and 120O. Values of A2 were estimated by applying a quadratic fit to the concentration dependence of Kc/Re. Alternatively, ( K c / R ~ was ) ~ /fit ~ according to Berry,18leading to somewhat higher A2 values. Because of the large experimental uncertainty (perhaps 50%) in A2, we do not put great emphasis on the absolute values, but the dependence on molecular weight is evident enough. As shown in Figure 3, all four PS samples exhibit a non-zero value for Az which increases significantly with decreasing molecular weight. Such behavior was predicted probably for the first time by Orofino and F l ~ r y who ,~~ used a model of polymeric coils with Gaussian segment distributions around their centers of mass. By taking into account three-body interactions within the overlap region of two neighboring polymers, they arrived at the relationship
+
Az = ( 1 6 a / 3 3 / 2 ) ( N , ( S 2 ) 3 / 2 / ~I)n [l X 1 ( d 2 / 4 ) + X2(7r”233/2/32)] (5) Here X1 and X2 are defined as
= ( 1 / 4 ) ( 3 / ~ ) 3 ’ 2 ( ~ 2 / V 1 N ~ ) ( ~ / [(1/2) ( S 2 ) 3-’ XI] 2) (64 Xz = 3(31’2/2~)3(~3/V1N~2)(1M3/(S2)3) [(1/3) - ~ 2 1 (6b) where u is the partial specific volume of the polymer, V1 the molar volume of the solvent, and x1 and x2 the interaction parameters characteristic of the polymer-solvent pair. As pointed out by Orofino and Flory, for high molecular weight samples in a good solvent, the influence of X2 on eq 5 becomes negligible because iV/ ( S2)3 is small in comparison to W/(S2)3/2; see eq 6a and 6b. In a 8solvent, however, x1approaches 1/2 and X1 diminishes, but not X2. If x2 is, as normally expected, considerably less than ll3,the term Xzwill become significant a t low molecular weights. Here the segment density within the overlap region increases, enhancing the number of contacts among three segments. This type of contact, which is not taken into account by xl,decreases the mutual interpenetration and leads to a nonvanishing second virial coefficient. At the €)-temperature,X1= 0, and a t low M the effect of chain stiffness will be to limit the three-segment term to the initial one in the expansion of the logarithm in eq 5. Under these conditions, we therefore have A2
- M / ( S 2 ) 3 / 2- MP
(low M , T = 0)
(7)
In the range lo3 < M < lo4, the wormlike model with the aforementioned parameters gives an exponent of about v = 0.7 for (S2)‘l2, so that in eq 7 we get p = -1.1, in adequate agreement with the cyclohexane data of Figure 3.
Conclusions (i) If the molecular weight is high enough, i.e., M > lo5 in the case of PS, mutually consistent relationships for \k(z) and a,(z) based on two-parameter theories can in principle reproduce experimental results. (ii) When z is modified by the molecular weight, curves of \ k ( z ) have a different shape from those obtained when z is modified by the temperature. Maximum values in plots of \k vs. CY deduced : from experimental data11J3are due to superposition of these two shapes. For this reason,
representations of @(as)where CY, is determined as a function of M should not be combined with plots where asis determined as a function of temperature. (iii) Two-parameter theories fail to describe completely the experimental behavior of short chains in both good and €)-solvents. However, a qualitatively correct interpretation is made possible if some third parameter is used: under good solvent conditions this can be achieved by taking into account chain stiffness and under €)-conditions by introducing a three-segment interaction parameter.
Acknowledgment. K.H. thanks the Alexander v. Humboldt Stiftung for the grant of a Feodor Lynen Fellowship, during the period of which part of this work was done. We thank the National Science Foundation for support under Grant DMR 8608633, Division of Materials Research, Polymers Program. The light-scattering experiments were performed a t the Institut fur Makromolekulare Chemie der Universitat Freiburg, FRG, under project SFB 60. Many helpful discussions with Prof. Walther Burchard during that time are much appreciated. Registry No. PS, 9003-53-6.
References and Notes Yamakawa, H. Modern Theory of Polymer Solutions; Harper & Row: New York, 1971; Section 21b. Reference 1, Section 20 and references therein. Reference 1, p 91. Reference 1; Section, 14, 15 and references therein. Muthukumar, M.; Nickel, B. J. Chem. Phys. 1984, 80, 5839. Gordon, M.; Ross-Murphy, S. B.; Suzuki, H. Eur. Polym. J. 1976, 12, 737. Suzuki, H. Macromolecules 1970, 3, 373. Burchard, W. Habilitationsschrift, Freiburg, 1966. Norisuye, T.; Kawahara, K.; Teramoto, A.; Fujita, H. J. Chem. Phys. 1968, 49, 4330. Takashima, K.; Nakae, K.; Shibata, M.; Yamakawa, H. Macromolecules 1974, 7, 641. Tanaka, G. J.Polym. Sci., Phys., Ed. 1979, 17, 305. Fujita, H.; Norisuye, T. Macromolecules 1985, 18, 1637. Suzuki, H. Macromolecules 1985, 18, 2082. Kurata, M.; Fukatsu, H.; Sotobayashi, H.; Yamakawa, H. J . Chem. Phys. 1964,41, 139. Yamakawa, H. J. Chem. Phys. 1968,48, 2103. Yamakawa, H.; Tanaka, G . J. Chem. Phys. 1967, 47, 3991. (a) Gobush, W.; Solc, K.; Stockmayer, W. H. J . Chem. Phys. 1974,60, 12. Here a slight maximum of q ( z ) as function of a, was calculated for ellipsoidally shaped self-intersecting chains. However, as in other approximations, Q decreases to 0 as a, approaches unity, as a consequence of the fact that \k was altered by varying the expansion of a relatively large chain, and chain stiffness was not considered. (b) Olaj, 0. F.; Zifferer, G., private communication. See also Zifferer, G. Doctoral Thesis, University of Vienna, 1982. Berry, G. C. J . Chem. Phys. 1966, 44, 4550. Miyaki, Y.; Einaga, Y.; Hirosye, T.; Fujita, H. Macromolecules 1977, 10, 1356. Miyaki, Y.; Einaga, Y.; Fujita, H. Macromolecule 1978, 11, 1180. Huber, K.; Burchard, W.; Akcasu, Z. A. Macromolecules 1985, 18, 2743. Varma, B. K.; Fujita, H.; Takahashi, M.; Nose, T. J . Polym. Sci., Polym. Phys. Ed. 1984, 22, 1718. Venkataswamy, K.; Jamieson, A. M.; Petschek, R. G. Macromolecules 1986, 19, 124. Oono, Y.; Freed, K. F. J. Phys. A: Math. Gen. 1982,15, 1931. Witten, T. A.; Schafer, L. J . Phys. A: Math. Gen. 1978, 11, 1843. Ragnetti, M.; Geiser, D.; Hijcker, H.; Oberthur, R. G. Makromol. Chem. 1985,186, 1701. Huber, K.; Bantle, S.; Lutz, P.; Burchard, W. Macromolecules 1985, 18, 1461. Benoit, H.; Doty, P. J. Phys. Chem. 1953, 57, 958. Zang, L.; Qui, D.; Qian, R. Polym. J . 1985, 17, 657. Yamakawa, H.; Stockmayer, W. H. J . Chem. Phys. 1972, 57, 2843. Zimm, B. J . Chem. Phys. 1946,14, 164. Orofino, T. A.; Flory, P. J . Chem. Phys. 1957, 26, 1067. Bantle, S.; Schmidt, M.; Burchard, W. Macromolecules 1982, 15, 1604.
